Dynamics of fractional quantum Hall Liquids with a pulse at the edge

Chen-Xin Jiang; Jie Li; Zi-Xiang Hu

REVIEW 3 major objections 7 minor 56 references

Dynamics of fractional quantum Hall Liquids with a pulse at the edge

T0 review · 3 major / 7 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A time-limited tip pulse near the edge of a $\nu=1/3$ Laughlin droplet sends the liquid into bulk magnetoroton excitations, with the edge-bulk mix controlled by pulse position.

desk verdict A useful new numerical experiment undermined by overclaimed edge propagation: the V1 model has zero edge velocity, so the chiral-edge claims rest on an unverified Coulomb assertion. read the letter →

arxiv 2507.10366 v1 pith:NFVVN6TC submitted 2025-07-14 cond-mat.mes-hall cond-mat.str-el

Jie Li , Chen-Xin Jiang , Zi-Xiang Hu This is my paper · ORCID

classification cond-mat.mes-hallcond-mat.str-el PACS 73.43.Lp71.10.Pm

keywords fractionalquantumHalleffectLaughlinstateedgemagnetoplasmonmagnetorotonquenchdynamicstippotentialexactdiagonalizationfidelityoscillations

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Starting from the $\nu=1/3$ Laughlin ground state in a disk, the paper applies a $\delta$-function tip potential near the edge for a finite duration, then removes it and follows the time evolution. It finds that the pulse excites both edge states and bulk states, and that the bulk part is dominated by neutral magnetoroton excitations rather than by generic quasiparticles. The central quantitative result is that the bulk contribution $S_{\mathrm{bulk}}$ peaks when the tip sits at the electron-density maximum $w\approx 5.4\,\ell_B$, while the edge contribution $S_{\mathrm{edge}}$ peaks at $w\approx 6.4\,\ell_B$, coinciding with the maximum of the edge dipole moment. These peak positions tie the edge-bulk balance of the pump-probe response to a geometric property of the edge, and the fidelity oscillations are shown to be governed by magnetoroton energy differences. The study matters because it offers a microscopic picture of how a localized voltage pulse at a fractional quantum Hall edge deposits energy into the gapped bulk, a process seen in recent pump-probe experiments.

What carries the argument

The machinery is exact time evolution of a lowest-Landau-level projected many-body Hamiltonian on a disk of $N_e=9$ electrons in $N_{\mathrm{orb}}=27$ orbitals, with the $V_1$ Haldane pseudopotential interaction whose densest zero-energy ground state is the Laughlin state. The pulse is a projected $\delta$-function potential $V(z)=U_\delta\,\delta(z-w)$ held for duration $\tau$. Two diagnostics carry the argument: the overlap sums $S_{\mathrm{bulk}}$ and $S_{\mathrm{edge}}$, which count how much of the post-pulse state lives in the bulk magnetoroton window versus the zero-energy edge sector, and the fidelity $f(t)=|\langle\Psi(0)|\Psi(t)\rangle|^2$, whose Fourier spectrum is compared directly with energy differences among the dominant excited states.

What would settle it

Repeat the exact-diagonalization pulse protocol with the Coulomb interaction, or on larger disks with $N_e=12,15$, and check whether the $S_{\mathrm{bulk}}$ maximum still coincides with the radial electron-density maximum and the $S_{\mathrm{edge}}$ maximum with the dipole-moment maximum, and whether the fidelity Fourier peaks still match the magnetoroton energy differences; a shift of the peak positions or a dominance of non-magnetoroton bulk states would contradict the central claim.

Watch

Extended reading notes

Core claim

The paper claims that a time-limited, spatially localized pulse at the edge of a $\nu=1/3$ Laughlin droplet acts as a pump that injects the liquid into two competing channels: edge excitations and bulk excitations. Within the $V_1$ model Hamiltonian, where edge states are exactly zero-energy and do not propagate along the boundary, the post-pulse state has significant overlap with the low-energy bulk states in the angular-momentum window $[M_0-N_e, M_0-1]$, and those states lie on the magnetoroton branch; the highest-overlap levels match the magnetoroton spectrum, and the energy differences among them reproduce the Fourier peaks of the time-dependent fidelity. The paper also establishes that the relative weight of the two channels is controlled by the pulse position: $S_{\mathrm{bulk}}$ is maximal at the radial electron-density peak, whereas $S_{\mathrm{edge}}$ is maximal where the edge dipole moment is maximal. For pulse strength and duration, the ground-state return amplitude oscillates as a two-level Rabi process with $\Delta U_\delta \cdot \tau = 2\pi$, allowing the excitation to be tuned by pulse shaping. The authors argue that the same qualitative behavior survives with Coulomb interaction, where the edge velocity becomes nonzero.

Load-bearing premise

The central calculation uses an idealized short-range interaction for which edge states have exactly zero energy and zero edge velocity, so the pulse cannot propagate along the boundary in the model; the paper asserts, in a single sentence, that the Coulomb interaction leaves the overall qualitative behavior similar.

Editorial extensions

If this is right

Positioning the excitation tip at the electron-density maximum selectively pumps the bulk magnetoroton branch, so the density disturbance diffuses inward from the edge on a timescale set by magnetoroton gaps.
Positioning the tip nearer the boundary selectively excites edge states, with maximum edge response at the dipole-moment maximum, so the edge-bulk mix is continuously tunable by pulse position.
Fourier analysis of the post-quench fidelity provides a dynamical spectroscopic route to magnetoroton energies: the oscillation frequencies equal energy differences between the dominant overlap states.
The Rabi-like relation $\Delta U_\delta\,\tau = 2\pi$ lets pulse duration and strength suppress or enhance the return to the ground state, giving a practical tuning knob for pump-probe experiments.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper leaves implicit that the coincidence between $S_{\mathrm{edge}}$ and the edge dipole moment suggests a general selection rule: a local probe couples most strongly to edge charge asymmetry; this could be tested at other fillings such as $\nu=2/3$ where upstream edge modes exist.
With Coulomb interaction the edge velocity becomes nonzero, so the chiral drift of the excited packet is expected to shift the apparent $S_{\mathrm{edge}}$ distribution in time; the $V_1$ result should be viewed as the zero-velocity limit of a family of edge-bulk dynamics.
A natural extension is to replace the square pulse by shaped pulses, such as Gaussian or chirped pulses, to selectively populate a single magnetoroton level, exploiting the two-level Rabi structure the paper identifies.
Finite-size scaling of $S_{\mathrm{bulk}}$ and $S_{\mathrm{edge}}$ on larger disks would show whether the peak positions track the density and dipole maxima universally or drift with $N_e$.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper does something genuinely new: it follows a time-limited tip pulse at the edge of a ν=1/3 Laughlin droplet and shows that the post-pulse state's composition depends on pulse position, with the bulk contribution peaking near the electron density maximum and the edge contribution peaking farther out. The fidelity analysis is internally consistent, and the match between Fourier peaks and energy differences from the same spectrum is a solid consistency check, though not an independent prediction. If the bulk-magnetoroton part is taken alone, the results are a reasonable numerical scenario.

The soft spots are real and load-bearing for the edge claims. The simulation uses the pure V1 interaction. As the authors themselves state in Section III, for V1 the edge states are zero-energy and the edge velocity is zero, so the impact of the tip potential does not propagate along the boundary. That means Figs. 2–5 contain no chiral edge propagation; the density changes are bulk diffusion only. The abstract and conclusions nevertheless claim excitations spread 'both along the edge and into the bulk' and 'align well with experimental observations.' That is not supported by the model actually simulated. The single sentence asserting that Coulomb interaction gives 'similar qualitative behavior' is not enough, especially when the experiments in Ref. 44 are about propagating chiral edge magnetoplasmons. This needs time-resolved density maps or at least Sbulk/Sedge curves for a Coulomb interaction before the edge-propagation half of the central claim is credible.

Also, Eqs. (5) and (6) are mathematically wrong as printed: they define the time-evolved state as an integral over time of the unitary evolution, which is not unitary. These look like typos, but they need fixing. The Sbulk definition is somewhat self-confirming: it sums overlaps only within the magnetoroton angular momentum window, so it labels those states as 'bulk' by construction. It is a reasonable operational choice, but it does not independently establish that those states are the dominant bulk excitations. The system is 9 electrons, and no code or data are released, so the numerics are not reproducible as is.

Who is this for? Researchers studying edge-bulk coupling in FQH systems, especially those connecting to the recent pump-probe experiments. The bulk-magnetoroton part is worth a referee's time, but the edge-propagation claim needs a real Coulomb calculation. I would send it to review with the expectation of major revision, not reject it outright.

Recommendation: engage with the paper, but only after the Coulomb calculation is provided and the printed errors are corrected.

Referee Report

3 major / 7 minor

Summary. The manuscript studies the quench dynamics of a ν=1/3 Laughlin droplet in a disk geometry under a time-limited delta-function tip potential. The authors consider 9 electrons in 27 orbitals, evolve the system under the V1 short-range interaction during and after the pulse, and analyze the residual density, the fidelity, and the overlaps of the post-pulse state with eigenstates of the static Hamiltonian. They introduce Sbulk and Sedge to quantify the bulk and edge contributions, and report that the bulk contribution peaks when the tip sits at the electron density maximum (w ≈ 5.4 lB), while the edge contribution peaks at w ≈ 6.4 lB, coinciding with the maximum of the edge dipole moment. They further attribute the bulk excitation predominantly to magnetoroton modes and interpret the pulse-strength/duration dependence as Rabi-like oscillations. The paper claims qualitative agreement with recent pump-probe experiments on chiral edge magnetoplasmons and bulk magnetorotons.

Significance. If the central claims hold, the paper provides a useful microscopic description of how a localized edge pulse couples to bulk and edge excitations in a fractional quantum Hall droplet. The main strengths are that the numerical study is parameter-free (no fitting to the data it explains), that the fidelity oscillation frequencies are internally consistent with energy differences extracted from the same spectrum, and that the pulse-position dependence of Sbulk and Sedge offers concrete, falsifiable predictions for future experiments. However, the significance is currently limited by two issues: the printed time-evolution formulas are non-unitary and the central edge-propagation claim rests on an unverified assertion about the Coulomb interaction. The bulk-magnetoroton part of the paper is likely sound and publishable, but the edge-chiral-transport half of the claim needs either additional calculation or a substantial reframing.

major comments (3)

[Section II, Eqs. (5) and (6)] The time evolution is printed as |Ψ(0)⟩ = ∫_0^τ dt exp(-iH2 t/ℏ)|Ψ1⟩ and |Ψ(t)⟩ = ∫_0^t dt′ exp(-iH1 t′/ℏ)|Ψ(0)⟩. These expressions are not unitary, do not conserve the norm, and have incorrect dimensions; the correct forms should be |Ψ(0)⟩ = exp(-iH2 τ/ℏ)|Ψ1⟩ and |Ψ(t)⟩ = exp(-iH1 t/ℏ)|Ψ(0)⟩. Because every subsequent quantity (density, fidelity, overlaps) is computed from these states, this is a load-bearing error. If the numerical code actually used the unitary propagators, the text must be corrected to match the code, and the authors should confirm that the reported results are unaffected.
[Section III, Eq. (7)] The residual density is defined as ρ(t) = ⟨Ψ(t)|Ψ(t)⟩ - ρ1. For a normalized many-body state, ⟨Ψ(t)|Ψ(t)⟩ = 1, so this expression cannot generate the spatially resolved density maps shown in Fig. 3. The definition should involve the electron density operator, e.g., ρ(r,t) = ⟨Ψ(t)|Σ_i δ(r-r_i)|Ψ(t)⟩ - ρ1(r). As printed, the paper's central observable is mathematically undefined, so this equation must be corrected before the results can be assessed.
[Section III, "Effect of Coulomb interaction", and Section V] The model actually simulated is the pure V1 hard-core interaction, and the paper explicitly states that for V1 the edge states are zero-energy eigenstates with zero edge velocity, so the tip impact does not propagate along the boundary. Yet the abstract and conclusions claim that excitations spread both along the edge and into the bulk, and that electrons move along the edge due to chiral edge modes. The only evidence that realistic Coulomb interaction changes this is a single paragraph asserting that the overall qualitative behavior remains similar, without showing any Coulomb-interaction density dynamics, Sbulk/Sedge curves, or edge-velocity data. Since the experiments in Refs. 44-46 concern propagating chiral edge magnetoplasmons, the edge-propagation half of the central claim is not supported by the presented simulation. The authors should either provide the Coulomb calculation or reframe the paper as a study of bulk diffusion and magnetoroton excitation in a zero-edge-velocity model.

minor comments (7)

[Fig. 3 caption] The caption states that the penetration depth is "significantly greater than that at w=7.0", which is self-referential; it should compare case B (w=7.0) with case A (w=5.4) or otherwise be reworded.
[Section IV heading] The heading "ANALYZE OF THE DETAILS OF THE TIP" should be corrected to "Analysis of the details of the tip".
[Throughout] The phrase "magnetic rotor" appears in the conclusions and elsewhere; this should be "magnetoroton".
[Section III, Eq. (9)] The text states that the magnetoroton subspace corresponds to angular momenta in the range [M0-Ne, M0], but the sum in Eq. (9) runs only up to M0-1. Please clarify the exact window used in the numerical calculation.
[Section III, "Effect of Coulomb interaction"] The claim about Coulomb interaction contains no quantitative data or figure. If this claim is retained, the authors should provide the system size, the relevant energy gaps, and a comparison of fidelity periods for the Coulomb case.
[Section II] The paper credits the time-dependent Lanczos algorithm in the acknowledgments but gives no numerical details (time step, truncation, convergence criteria). A brief description of the numerical integration would help reproducibility.
[Fig. 5(b) and Table I] The frequency peaks in Fig. 5(b) are read from the same eigenstates and energies that define the overlaps in Fig. 4, so the agreement in Table I is an internal consistency check rather than an independent prediction; this should be stated explicitly.

Circularity Check

1 steps flagged · score 2.0 of 10

No significant circularity: the fidelity-frequency match is tautological via Eq. (8), but the position-dependent Sbulk/Sedge and magnetoroton-weight results are independent exact-diagonalization computations; V1 zero-edge-velocity is an external-validity caveat.

self definitional [Section III, Eq. (8) and Fig. 5(b)/Table I]
"Analytically, the fidelity f(t) can be expanded in terms of energy eigenstates as: |⟨Ψ(0)|Ψ(t)⟩|^2 = |∑_i |c_i|^2 e^{-iE_i t/ℏ}|^2 = ∑_i |c_i|^4 + ∑_{i<j} 2|c_i|^2|c_j|^2 cos((E_i − E_j)t/ℏ). (8) ... The spectral peaks in F(Ω) exhibit a correspondence with the energy differences between these levels, as detailed in Table I."

This confirmation reduces by construction: the fidelity of a state evolving under H1 has Fourier components exactly at the eigenenergy differences E_i − E_j by Eq. (8), using the same eigenstates and energies read off the same spectrum. The agreement between the Fourier peaks of f(t) and the levels labeled in Fig. 4/Table I is therefore an identity, not an independent dynamical prediction of the magnetoroton spectrum. It validates internal numerical consistency but does not constitute evidence that magnetorotons govern the dynamics; the latter claim rests on the overlap weights, which are an independent computation.

full rationale

The paper's central content — the residual-density maps, the overlap weights in Fig. 4, the Sbulk(w) peak at the density maximum, and the Sedge(w) peak at the dipole-moment maximum — is obtained by exact diagonalization and time propagation of the stated V1 Hamiltonian with no fitted parameters, so those results are not circular. The use of 'our earlier research [52]' to place the magnetoroton sector in the [M0−Ne, M0] window is a legitimate citation of a prior, externally published calculation, not an imported conclusion that contains the present result; the high-overlap states are identified by actual computed overlaps. The only circular element I found is the fidelity-frequency matching: Eq. (8) makes the Fourier peaks equal to eigenenergy differences by definition, so presenting the Table I match as 'confirming' magnetoroton control of the oscillations is a self-consistency check rather than an independent prediction. The manuscript itself flags a separate non-circular limitation, in Section III: 'For model Hamiltonian with V1 interaction, the edge states are also zero energy eigenstates, and thus the edge velocity is zero. In this scenario, the impact of the tip potential does not propagate along the boundary,' and the subsequent Coulomb assertion ('the overall qualitative behavior remains similar') is presented without supporting data; I treat that as an external-validity caveat, not as circularity. Overall score 2: one minor self-definitional consistency check, with the central claims retaining independent computational content.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The model relies on the standard Laughlin/Haldane pseudopotential framework, the definition of edge states as zero-energy modes of the V1 interaction, and the identification of the magnetoroton sector from the authors' prior work. The analysis introduces two manual choices: the angular momentum window and the number of states summed in Sbulk. No fitted parameters appear; U_delta and tau are simulation controls.

free parameters (4)

Pulse strength U_delta = 1 (main runs), scanned up to U_delta/pi=4 in Fig. 8
Simulation parameter controlling tip potential amplitude; not fitted to data.
Pulse duration tau = 3 (main runs), scanned in Fig. 8
Simulation parameter controlling pulse duration; inverse relation with Rabi period observed.
Angular momentum window for Sbulk = [M0-Ne, M0-1] = [99,108] for Ne=9
Post-hoc selection of the sector where magnetorotons live; defines Sbulk and thus the claim of bulk predominance.
Lowest 40 excited states per sector in Eq. (9) = 40
Truncation for Sbulk; affects the value of Sbulk.

assumptions (4)

standard math The Laughlin state at nu=1/3 is the densest zero-energy eigenstate of the V1 hard-core interaction (Refs. 2,48).
Basis for the model Hamiltonian H1.
domain assumption Edge states are zero-energy eigenstates for the V1 model.
Used to define Sedge via energy gap; paper notes edge velocity is zero in this model.
domain assumption The magnetoroton subspace is the zero-COM angular momentum sector with M in [M0-Ne, M0-1] (Ref. 52, by same group).
Used to define Sbulk and to interpret the dominant bulk excitations.
standard math Time evolution is generated by unitary operators exp(-iHt/hbar); Eqs. (5)-(6) as printed are inconsistent with this.
The paper's printed evolution equations use integrals instead of exponentials; results imply the exponentials were implemented.

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Cite this review

Pith. "Pith review of Dynamics of fractional quantum Hall Liquids with a pulse at the edge." pith.science (2026). https://pith.science/paper/NFVVN6TC

@misc{pith2026250710366,
  author       = {Pith},
  title        = {Pith review of: Dynamics of fractional quantum Hall Liquids with a pulse at the edge},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NFVVN6TC}},
  note         = {Machine review of arXiv:2507.10366}
}

read the original abstract

Motivated by recent experimental advancements in scanning optical stroboscopic confocal microscopy and spectroscopy measurements, which have facilitated exceptional energy-space-time resolution for investigating edge and bulk dynamics in fractional quantum Hall systems, we formulated a model for the pump-probe process on the edge. Starting with a ground state, we applied a tip potential near the fractional quantum Hall liquid edge, which was subsequently turned off after a defined time duration. By examining how the specific nature of the tip potential influences the evolution of the wave function and its distribution in energy spectrum, we identify that quench dynamics of the edge pulse leads to excitations that spread both along the edge and perpendicularly into the bulk. Moreover, magnetoroton excitations are predominant among the bulk excitations. These results align well with the experimental observations. Furthermore, we analyzed the effects of the tip's position, intensity, and duration on the dynamics.

Figures

Figures reproduced from arXiv: 2507.10366 by the authors.

**Figure 2.** FIG. 2: The radial density profile of the Laughlin state for [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3: Evolution of the residual density under quenched [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: FIG. 4: Energy spectrum for 9 electrons and 27 orbitals. [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

**Figure 5.** Figure 5: FIG. 5: In a 9-electron 27-orbital system with [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

**Figure 6.** Figure 6: FIG. 6: The dependence of [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]

**Figure 7.** Figure 7: FIG. 7: The pulse position dependence of [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]

**Figure 8.** Figure 8: FIG. 8: Variations of the wavefunction overlap [PITH_FULL_IMAGE:figures/full_fig_p007_8.png]

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

56 extracted references · 34 canonical work pages

[1]

author D. C. Tsui , author H. L. Stormer , and author A. C. Gossard , journal Phys. Rev. Lett. volume 48 , pages 1559 ( year 1982 ), ://link.aps.org/doi/10.1103/PhysRevLett.48.1559

doi:10.1103/physrevlett.48.1559 1982
[2]

author R. B. Laughlin , journal Phys. Rev. Lett. volume 50 , pages 1395 ( year 1983 ), ://link.aps.org/doi/10.1103/PhysRevLett.50.1395

doi:10.1103/physrevlett.50.1395 1983
[3]

Wilczek , journal Phys

author F. Wilczek , journal Phys. Rev. Lett. volume 49 , pages 957 ( year 1982 ), ://link.aps.org/doi/10.1103/PhysRevLett.49.957

doi:10.1103/physrevlett.49.957 1982
[4]

author B. I. Halperin , journal Phys. Rev. Lett. volume 52 , pages 1583 ( year 1984 ), ://link.aps.org/doi/10.1103/PhysRevLett.52.1583

doi:10.1103/physrevlett.52.1583 1984
[5]

Arovas , author J

author D. Arovas , author J. R. Schrieffer , and author F. Wilczek , journal Phys. Rev. Lett. volume 53 , pages 722 ( year 1984 ), ://link.aps.org/doi/10.1103/PhysRevLett.53.722

doi:10.1103/physrevlett.53.722 1984
[6]

author A. Y. Kitaev , journal Annals of Physics volume 303 , pages 2 ( year 2003 ), ://www.sciencedirect.com/science/article/pii/S0003491602000180

work page 2003
[7]

Nayak , author S

author C. Nayak , author S. H. Simon , author A. Stern , author M. Freedman , and author S. Das Sarma , journal Rev. Mod. Phys. volume 80 , pages 1083 ( year 2008 ), ://link.aps.org/doi/10.1103/RevModPhys.80.1083

doi:10.1103/revmodphys.80.1083 2008
[8]

author X. G. Wen , journal International Journal of Modern Physics B volume 6 , pages 1711 ( year 1992 ), ://doi.org/10.1142/S0217979292000840

doi:10.1142/s0217979292000840 1992

Show all 56 references

[9]

author X. G. Wen , journal Phys. Rev. B volume 41 , pages 12838 ( year 1990 ), ://link.aps.org/doi/10.1103/PhysRevB.41.12838

1990 doi
[10]

author S. M. Girvin , author A. H. MacDonald , and author P. M. Platzman , journal Phys. Rev. Lett. volume 54 , pages 581 ( year 1985 ), ://link.aps.org/doi/10.1103/PhysRevLett.54.581

1985 doi
[11]

author S. M. Girvin , author A. H. MacDonald , and author P. M. Platzman , journal Phys. Rev. B volume 33 , pages 2481 ( year 1986 ), ://link.aps.org/doi/10.1103/PhysRevB.33.2481

1986 doi
[12]

Yang , author Z.-X

author B. Yang , author Z.-X. Hu , author Z. Papi c \' c , and author F. D. M. Haldane , journal Phys. Rev. Lett. volume 108 , pages 256807 ( year 2012 ), ://link.aps.org/doi/10.1103/PhysRevLett.108.256807

2012 doi
[13]

Chandran , author M

author A. Chandran , author M. Hermanns , author N. Regnault , and author B. A. Bernevig , journal Phys. Rev. B volume 84 , pages 205136 ( year 2011 ), ://link.aps.org/doi/10.1103/PhysRevB.84.205136

2011 doi
[14]

Luo , author B

author Z.-X. Luo , author B. G. Pankovich , author Y. Hu , and author Y.-S. Wu , journal Phys. Rev. B volume 99 , pages 205137 ( year 2019 ), ://link.aps.org/doi/10.1103/PhysRevB.99.205137

2019 doi
[15]

Sahasrabudhe , author B

author H. Sahasrabudhe , author B. Novakovic , author J. Nakamura , author S. Fallahi , author M. Povolotskyi , author G. Klimeck , author R. Rahman , and author M. J. Manfra , journal Phys. Rev. B volume 97 , pages 085302 ( year 2018 ), ://link.aps.org/doi/10.1103/PhysRevB.97.085302

2018 doi
[16]

Ji , author Y

author Y. Ji , author Y. Chung , author D. Sprinzak , author M. Heiblum , author D. Mahalu , and author H. Shtrikman , journal Nature volume 422 , pages 415 ( year 2003 ), ://doi.org/10.1038/nature01503

2003 doi
[17]

Nakamura , author S

author J. Nakamura , author S. Fallahi , author H. Sahasrabudhe , author R. Rahman , author S. Liang , author G. C. Gardner , and author M. J. Manfra , journal Nature Physics volume 15 , pages 563 ( year 2019 ), ://doi.org/10.1038/s41567-019-0441-8

2019 doi
[18]

Nakamura , author S

author J. Nakamura , author S. Liang , author G. C. Gardner , and author M. J. Manfra , journal Nature Physics volume 16 , pages 931 ( year 2020 ), ://doi.org/10.1038/s41567-020-1019-1

2020 doi
[19]

Bartolomei , author M

author H. Bartolomei , author M. Kumar , author R. Bisognin , author A. Marguerite , author J.-M. Berroir , author E. Bocquillon , author B. Plaçais , author A. Cavanna , author Q. Dong , author U. Gennser , et al. , journal Science volume 368 , pages 173 ( year 2020 ), ://www...

2020 doi
[20]

author A. H. MacDonald , journal Phys. Rev. Lett. volume 64 , pages 220 ( year 1990 ), ://link.aps.org/doi/10.1103/PhysRevLett.64.220

1990 doi
[21]

author C. d. C. Chamon and author X. G. Wen , journal Phys. Rev. B volume 49 , pages 8227 ( year 1994 ), ://link.aps.org/doi/10.1103/PhysRevB.49.8227

1994 doi
[22]

Wan , author E

author X. Wan , author E. H. Rezayi , and author K. Yang , journal Phys. Rev. B volume 68 , pages 125307 ( year 2003 ), ://link.aps.org/doi/10.1103/PhysRevB.68.125307

2003 doi
[23]

Wan , author K

author X. Wan , author K. Yang , and author E. H. Rezayi , journal Phys. Rev. Lett. volume 88 , pages 056802 ( year 2002 ), ://link.aps.org/doi/10.1103/PhysRevLett.88.056802

2002 doi
[24]

Hu , author R

author Z.-X. Hu , author R. N. Bhatt , author X. Wan , and author K. Yang , journal Phys. Rev. Lett. volume 107 , pages 236806 ( year 2011 ), ://link.aps.org/doi/10.1103/PhysRevLett.107.236806

2011 doi
[25]

Sabo , author I

author R. Sabo , author I. Gurman , author A. Rosenblatt , author F. Lafont , author D. Banitt , author J. Park , author M. Heiblum , author Y. Gefen , author V. Umansky , and author D. Mahalu , journal Nature Physics volume 13 , pages 491 ( year 2017 ), ://doi.org/10.1038/nphys4010

2017 doi
[26]

author C. L. Kane and author M. P. A. Fisher , journal Phys. Rev. B volume 46 , pages 15233 ( year 1992 ), ://link.aps.org/doi/10.1103/PhysRevB.46.15233

1992 doi
[27]

author C. de C. Chamon and author X. G. Wen , journal Phys. Rev. Lett. volume 70 , pages 2605 ( year 1993 ), ://link.aps.org/doi/10.1103/PhysRevLett.70.2605

1993 doi
[28]

author C. L. Kane and author M. P. A. Fisher , journal Phys. Rev. Lett. volume 72 , pages 724 ( year 1994 ), ://link.aps.org/doi/10.1103/PhysRevLett.72.724

1994 doi
[29]

Moon , author H

author K. Moon , author H. Yi , author C. L. Kane , author S. M. Girvin , and author M. P. A. Fisher , journal Phys. Rev. Lett. volume 71 , pages 4381 ( year 1993 ), ://link.aps.org/doi/10.1103/PhysRevLett.71.4381

1993 doi
[30]

Lin , author C

author X. Lin , author C. Dillard , author M. A. Kastner , author L. N. Pfeiffer , and author K. W. West , journal Phys. Rev. B volume 85 , pages 165321 ( year 2012 ), ://link.aps.org/doi/10.1103/PhysRevB.85.165321

2012 doi
[31]

Bid , author N

author A. Bid , author N. Ofek , author H. Inoue , author M. Heiblum , author C. L. Kane , author V. Umansky , and author D. Mahalu , journal Nature volume 466 , pages 585 ( year 2010 ), ://doi.org/10.1038/nature09277

2010 doi
[32]

Wassermeier , author J

author M. Wassermeier , author J. Oshinowo , author J. P. Kotthaus , author A. H. MacDonald , author C. T. Foxon , and author J. J. Harris , journal Phys. Rev. B volume 41 , pages 10287 ( year 1990 ), ://link.aps.org/doi/10.1103/PhysRevB.41.10287

1990 doi
[33]

author R. C. Ashoori , author H. L. Stormer , author L. N. Pfeiffer , author K. W. Baldwin , and author K. West , journal Phys. Rev. B volume 45 , pages 3894 ( year 1992 ), ://link.aps.org/doi/10.1103/PhysRevB.45.3894

1992 doi
[34]

author N. B. Zhitenev , author R. J. Haug , author K. v. Klitzing , and author K. Eberl , journal Phys. Rev. Lett. volume 71 , pages 2292 ( year 1993 ), ://link.aps.org/doi/10.1103/PhysRevLett.71.2292

1993 doi
[35]

author I. L. Aleiner and author L. I. Glazman , journal Phys. Rev. Lett. volume 72 , pages 2935 ( year 1994 ), ://link.aps.org/doi/10.1103/PhysRevLett.72.2935

1994 doi
[36]

Ernst , author N

author G. Ernst , author N. B. Zhitenev , author R. J. Haug , and author K. von Klitzing , journal Phys. Rev. Lett. volume 79 , pages 3748 ( year 1997 ), ://link.aps.org/doi/10.1103/PhysRevLett.79.3748

1997 doi
[38]

Bhattacharyya , author M

author R. Bhattacharyya , author M. Banerjee , author M. Heiblum , author D. Mahalu , and author V. Umansky , journal Phys. Rev. Lett. volume 122 , pages 246801 ( year 2019 ), ://link.aps.org/doi/10.1103/PhysRevLett.122.246801

2019 doi
[39]

Venkatachalam , author S

author V. Venkatachalam , author S. Hart , author L. Pfeiffer , author K. West , and author A. Yacoby , journal Nature Physics volume 8 , pages 676 ( year 2012 ), ://doi.org/10.1038/nphys2384

2012 doi
[40]

Banerjee , author M

author M. Banerjee , author M. Heiblum , author V. Umansky , author D. E. Feldman , author Y. Oreg , and author A. Stern , journal Nature volume 559 , pages 205 ( year 2018 ), ://doi.org/10.1038/s41586-018-0184-1

2018 doi
[41]

Banerjee , author M

author M. Banerjee , author M. Heiblum , author A. Rosenblatt , author Y. Oreg , author D. E. Feldman , author A. Stern , and author V. Umansky , journal Nature volume 545 , pages 75 ( year 2017 ), ://doi.org/10.1038/nature22052

2017 doi
[42]

author R. A. Melcer , author A. Gil , author A. K. Paul , author P. Tiwari , author V. Umansky , author M. Heiblum , author Y. Oreg , author A. Stern , and author E. Berg , journal Nature volume 625 , pages 489 ( year 2024 ), ://doi.org/10.1038/s41586-023-06858-z

2024 doi
[43]

Hayakawa , author K

author J. Hayakawa , author K. Muraki , and author G. Yusa , journal Nature Nanotechnology volume 8 , pages 31 ( year 2013 ), ://doi.org/10.1038/nnano.2012.209

2013 doi
[44]

Kamiyama , author M

author A. Kamiyama , author M. Matsuura , author J. N. Moore , author T. Mano , author N. Shibata , and author G. Yusa , journal Phys. Rev. Res. volume 4 , pages L012040 ( year 2022 ), ://link.aps.org/doi/10.1103/PhysRevResearch.4.L012040

2022 doi
[45]

Kamiyama , author M

author A. Kamiyama , author M. Matsuura , author J. N. Moore , author T. Mano , author N. Shibata , and author G. Yusa , journal Applied Physics Letters volume 122 , pages 202103 ( year 2023 ), ://doi.org/10.1063/5.0138332

2023 doi
[46]

France , author Y

author Q. France , author Y. Jeong , author A. Kamiyama , author T. Mano , author K. ichi Sasaki , author M. Hotta , and author G. Yusa ( year 2025 ), arXiv:2502.01052

2025 arXiv
[47]

Hu , author X

author Z.-X. Hu , author X. Wan , and author P. Schmitteckert , journal Phys. Rev. B volume 77 , pages 075331 ( year 2008 ), ://link.aps.org/doi/10.1103/PhysRevB.77.075331

2008 doi
[48]

author F. D. M. Haldane , journal Phys. Rev. Lett. volume 51 , pages 605 ( year 1983 ), ://link.aps.org/doi/10.1103/PhysRevLett.51.605

1983 doi
[49]

Park and author F

author Y. Park and author F. D. M. Haldane , journal Phys. Rev. B volume 90 , pages 045123 ( year 2014 ), ://link.aps.org/doi/10.1103/PhysRevB.90.045123

2014 doi
[50]

Yang and author Z.-X

author Y. Yang and author Z.-X. Hu , journal Phys. Rev. B volume 107 , pages 115162 ( year 2023 ), ://link.aps.org/doi/10.1103/PhysRevB.107.115162

2023 doi
[51]

Hu , author E

author Z.-X. Hu , author E. H. Rezayi , author X. Wan , and author K. Yang , journal Phys. Rev. B volume 80 , pages 235330 ( year 2009 b ), ://link.aps.org/doi/10.1103/PhysRevB.80.235330

2009 doi
[52]

Yang , author Q

author W.-Q. Yang , author Q. Li , author L.-P. Yang , and author Z.-X. Hu , journal Chinese Physics B volume 28 , eid 067303 ( year 2019 ), ://cpb.iphy.ac.cn/EN/abstract/article_121668.shtml

2019
[53]

Yang , author S

author Y. Yang , author S. Pu , author Y. Hu , and author Z.-X. Hu , journal Phys. Rev. B volume 111 , pages 195139 ( year 2025 ), ://link.aps.org/doi/10.1103/PhysRevB.111.195139

2025 doi
[54]

Wu , author B

author Y.-L. Wu , author B. Estienne , author N. Regnault , and author B. A. Bernevig , journal Phys. Rev. Lett. volume 113 , pages 116801 ( year 2014 ), ://link.aps.org/doi/10.1103/PhysRevLett.113.116801

2014 doi
[55]

Li , author N

author Q. Li , author N. Jiang , author Z. Zhu , and author Z.-X. Hu , journal New Journal of Physics volume 17 , pages 095006 ( year 2015 ), ://dx.doi.org/10.1088/1367-2630/17/9/095006

2015 doi
[56]

Liu , author R

author Z. Liu , author R. N. Bhatt , and author N. Regnault , journal Phys. Rev. B volume 91 , pages 045126 ( year 2015 ), ://link.aps.org/doi/10.1103/PhysRevB.91.045126

2015 doi
[57]

Li , author D

author J. Li , author D. Ye , author C.-X. Jiang , author N. Jiang , author X. Wan , and author Z.-X. Hu , journal Phys. Rev. B volume 105 , pages 195311 ( year 2022 ), ://link.aps.org/doi/10.1103/PhysRevB.105.195311

2022 doi

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

author D. C. Tsui , author H. L. Stormer , and author A. C. Gossard , journal Phys. Rev. Lett. volume 48 , pages 1559 ( year 1982 ), ://link.aps.org/doi/10.1103/PhysRevLett.48.1559

doi:10.1103/physrevlett.48.1559 1982

[2] [2]

author R. B. Laughlin , journal Phys. Rev. Lett. volume 50 , pages 1395 ( year 1983 ), ://link.aps.org/doi/10.1103/PhysRevLett.50.1395

doi:10.1103/physrevlett.50.1395 1983

[3] [3]

Wilczek , journal Phys

author F. Wilczek , journal Phys. Rev. Lett. volume 49 , pages 957 ( year 1982 ), ://link.aps.org/doi/10.1103/PhysRevLett.49.957

doi:10.1103/physrevlett.49.957 1982

[4] [4]

author B. I. Halperin , journal Phys. Rev. Lett. volume 52 , pages 1583 ( year 1984 ), ://link.aps.org/doi/10.1103/PhysRevLett.52.1583

doi:10.1103/physrevlett.52.1583 1984

[5] [5]

Arovas , author J

author D. Arovas , author J. R. Schrieffer , and author F. Wilczek , journal Phys. Rev. Lett. volume 53 , pages 722 ( year 1984 ), ://link.aps.org/doi/10.1103/PhysRevLett.53.722

doi:10.1103/physrevlett.53.722 1984

[6] [6]

author A. Y. Kitaev , journal Annals of Physics volume 303 , pages 2 ( year 2003 ), ://www.sciencedirect.com/science/article/pii/S0003491602000180

work page 2003

[7] [7]

Nayak , author S

author C. Nayak , author S. H. Simon , author A. Stern , author M. Freedman , and author S. Das Sarma , journal Rev. Mod. Phys. volume 80 , pages 1083 ( year 2008 ), ://link.aps.org/doi/10.1103/RevModPhys.80.1083

doi:10.1103/revmodphys.80.1083 2008

[8] [8]

author X. G. Wen , journal International Journal of Modern Physics B volume 6 , pages 1711 ( year 1992 ), ://doi.org/10.1142/S0217979292000840

doi:10.1142/s0217979292000840 1992